p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.46C23, C4.562+ 1+4, C8⋊8D4⋊50C2, C8⋊9D4⋊14C2, C4⋊C8⋊34C22, C4⋊C4.365D4, C4⋊Q8⋊23C22, D4⋊2Q8⋊18C2, (C2×D4).169D4, D4⋊5D4.3C2, C8.D4⋊23C2, (C4×Q8)⋊27C22, C4.Q8⋊25C22, C8⋊C4⋊23C22, C2.D8⋊36C22, D4.23(C4○D4), C22⋊SD16⋊21C2, D4.7D4⋊42C2, C4⋊C4.408C23, C22⋊C8⋊30C22, (C2×C4).503C24, (C2×C8).353C23, Q8.D4⋊40C2, C22⋊C4.165D4, (C22×C8)⋊43C22, (C2×Q16)⋊32C22, C23.322(C2×D4), C22⋊Q8⋊18C22, SD16⋊C4⋊34C2, C2.75(D4○SD16), Q8⋊C4⋊43C22, (C4×D4).156C22, (C2×D4).232C23, C4⋊D4.82C22, (C2×Q8).216C23, C2.139(D4⋊5D4), C42⋊C2⋊23C22, C23.19D4⋊32C2, C23.24D4⋊31C2, C23.37D4⋊13C2, C23.20D4⋊32C2, (C2×SD16).54C22, C4.4D4.64C22, C22.763(C22×D4), D4⋊C4.187C22, C2.84(D8⋊C22), C22.50C24⋊4C2, (C22×C4).1147C23, (C22×D4).411C22, C42.28C22⋊14C2, (C2×M4(2)).111C22, C4.228(C2×C4○D4), (C2×C4).600(C2×D4), (C2×C4○D4).208C22, SmallGroup(128,2043)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.46C23
G = < a,b,c,d,e | a4=b4=d2=1, c2=a2, e2=b2, ab=ba, cac-1=eae-1=a-1, dad=ab2, cbc-1=dbd=b-1, be=eb, dcd=bc, ece-1=a2c, ede-1=b2d >
Subgroups: 432 in 201 conjugacy classes, 86 normal (84 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C24, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C22⋊C4, C42⋊C2, C4×D4, C4×Q8, C4×Q8, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42⋊2C2, C4⋊Q8, C22×C8, C2×M4(2), C2×SD16, C2×Q16, C22×D4, C2×C4○D4, C23.24D4, C23.37D4, C8⋊9D4, SD16⋊C4, C22⋊SD16, D4.7D4, Q8.D4, C8⋊8D4, C8.D4, D4⋊2Q8, C23.19D4, C23.20D4, C42.28C22, D4⋊5D4, C22.50C24, C42.46C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, D4⋊5D4, D8⋊C22, D4○SD16, C42.46C23
Character table of C42.46C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ9 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ10 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ11 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ12 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ13 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ14 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ15 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ16 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ17 | 2 | 2 | 2 | 2 | 0 | 2 | 0 | -2 | 0 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 0 | 2 | 0 | 2 | 0 | -2 | -2 | -2 | -2 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 0 | -2 | 0 | -2 | 0 | -2 | 2 | 2 | -2 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 0 | -2 | 0 | 2 | 0 | -2 | 2 | 2 | -2 | -2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | -2 | 2 | 0 | -2 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | -2 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | -2 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 2 | 0 | -2 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ26 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
| ρ27 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
| ρ28 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√-2 | 0 | 2√-2 | 0 | 0 | 0 | complex lifted from D4○SD16 |
| ρ29 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√-2 | 0 | -2√-2 | 0 | 0 | 0 | complex lifted from D4○SD16 |
►On 32 points
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)
(1 25 18 22)(2 26 19 23)(3 27 20 24)(4 28 17 21)(5 10 29 16)(6 11 30 13)(7 12 31 14)(8 9 32 15)
(1 9 3 11)(2 12 4 10)(5 26 7 28)(6 25 8 27)(13 18 15 20)(14 17 16 19)(21 29 23 31)(22 32 24 30)
(1 3)(2 17)(4 19)(5 14)(6 9)(7 16)(8 11)(10 31)(12 29)(13 32)(15 30)(18 20)(21 23)(22 27)(24 25)(26 28)
(1 22 18 25)(2 21 19 28)(3 24 20 27)(4 23 17 26)(5 10 29 16)(6 9 30 15)(7 12 31 14)(8 11 32 13)
G:=sub<Sym(32)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,25,18,22)(2,26,19,23)(3,27,20,24)(4,28,17,21)(5,10,29,16)(6,11,30,13)(7,12,31,14)(8,9,32,15), (1,9,3,11)(2,12,4,10)(5,26,7,28)(6,25,8,27)(13,18,15,20)(14,17,16,19)(21,29,23,31)(22,32,24,30), (1,3)(2,17)(4,19)(5,14)(6,9)(7,16)(8,11)(10,31)(12,29)(13,32)(15,30)(18,20)(21,23)(22,27)(24,25)(26,28), (1,22,18,25)(2,21,19,28)(3,24,20,27)(4,23,17,26)(5,10,29,16)(6,9,30,15)(7,12,31,14)(8,11,32,13)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,25,18,22)(2,26,19,23)(3,27,20,24)(4,28,17,21)(5,10,29,16)(6,11,30,13)(7,12,31,14)(8,9,32,15), (1,9,3,11)(2,12,4,10)(5,26,7,28)(6,25,8,27)(13,18,15,20)(14,17,16,19)(21,29,23,31)(22,32,24,30), (1,3)(2,17)(4,19)(5,14)(6,9)(7,16)(8,11)(10,31)(12,29)(13,32)(15,30)(18,20)(21,23)(22,27)(24,25)(26,28), (1,22,18,25)(2,21,19,28)(3,24,20,27)(4,23,17,26)(5,10,29,16)(6,9,30,15)(7,12,31,14)(8,11,32,13) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32)], [(1,25,18,22),(2,26,19,23),(3,27,20,24),(4,28,17,21),(5,10,29,16),(6,11,30,13),(7,12,31,14),(8,9,32,15)], [(1,9,3,11),(2,12,4,10),(5,26,7,28),(6,25,8,27),(13,18,15,20),(14,17,16,19),(21,29,23,31),(22,32,24,30)], [(1,3),(2,17),(4,19),(5,14),(6,9),(7,16),(8,11),(10,31),(12,29),(13,32),(15,30),(18,20),(21,23),(22,27),(24,25),(26,28)], [(1,22,18,25),(2,21,19,28),(3,24,20,27),(4,23,17,26),(5,10,29,16),(6,9,30,15),(7,12,31,14),(8,11,32,13)]])
Matrix representation of C42.46C23 ►in GL6(𝔽17)
| 13 | 0 | 0 | 0 | 0 | 0 |
| 2 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 4 | 0 | 0 | 0 | 0 |
| 8 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 1 | 4 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(17))| [13,2,0,0,0,0,0,4,0,0,0,0,0,0,0,13,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,0,0,0,4,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[1,8,0,0,0,0,4,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,16,0],[1,0,0,0,0,0,4,16,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0] >;
C42.46C23 in GAP, Magma, Sage, TeX
C_4^2._{46}C_2^3 % in TeX
G:=Group("C4^2.46C2^3"); // GroupNames label
G:=SmallGroup(128,2043);
// by ID
G=gap.SmallGroup(128,2043);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,456,758,723,346,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=d^2=1,c^2=a^2,e^2=b^2,a*b=b*a,c*a*c^-1=e*a*e^-1=a^-1,d*a*d=a*b^2,c*b*c^-1=d*b*d=b^-1,b*e=e*b,d*c*d=b*c,e*c*e^-1=a^2*c,e*d*e^-1=b^2*d>;
// generators/relations
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